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Chapter 8: Problem 18

A sample of methane \(\left(\mathrm{CH}_{4}\right)\) has a volume of \(25 \mathrm{~mL}\) at a pressure of 0.80 atm. What is the final volume, in milliliters, of the gas at each of the following pressures, if there is no change in temperature and amount of gas? a. \(0.40 \mathrm{~atm}\) b. \(2.00 \mathrm{~atm}\) c. \(2500 \mathrm{mmHg}\) d. 80.0 Torr

Short Answer

Expert verified
50 mL, 10 mL, 6.08 mL, 190.5 mL

Step by step solution

01

Identify Given Values

The initial volume (\(V_1\)) is 25 mL and the initial pressure (\(P_1\)) is 0.80 atm. We need to find the final volume (\(V_2\)) for each given final pressure (\(P_2\)).
02

Recall Boyle's Law

Boyle's Law states that, for a constant temperature and amount of gas, the product of pressure and volume is constant: \(P_1 \cdot V_1 = P_2 \cdot V_2\). Rearrange the equation to find the final volume: \(V_2 = \frac{P_1 \cdot V_1}{P_2}\).
03

Solve for Given Pressure 0.40 atm

Using \(P_2 = 0.40 \space atm\), plug the values into Boyle's Law equation: \[V_2 = \frac{0.80 \times 25}{0.40} = 50 \space mL\].
04

Solve for Given Pressure 2.00 atm

Using \(P_2 = 2.00 \space atm\), plug the values into Boyle's Law equation: \[V_2 = \frac{0.80 \times 25}{2.00} = 10 \space mL\].
05

Convert mmHg to atm and Solve

Convert \(P_2 = 2500 \space mmHg\) to atm using the conversion factor (1 atm = 760 mmHg): \[P_2 = \frac{2500}{760} \approx 3.29 \space atm\]. Next, use Boyle's Law: \[V_2 = \frac{0.80 \times 25}{3.29} \approx 6.08 \space mL\].
06

Convert Torr to atm and Solve

Convert \(P_2 = 80.0 \space Torr\) to atm using the conversion factor (1 atm = 760 Torr): \[P_2 = \frac{80}{760} = 0.105 \space atm\]. Next, use Boyle's Law: \[V_2 = \frac{0.80 \times 25}{0.105} \approx 190.5 \space mL\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

gas laws
Gas laws are rules that describe how gases behave. In particular, they show how volume, temperature, and pressure are related for a fixed amount of gas. One important gas law is Boyle's Law. Boyle's Law explains how the pressure and volume of a gas are related when the temperature remains constant. If the volume of gas decreases, the pressure increases, and vice versa. By understanding gas laws, we can predict how a gas will behave under different conditions. This is useful for many practical applications, from inflating tires to understanding natural processes, like how our lungs work.
pressure-volume relationship
The pressure-volume relationship is crucial for understanding how gases behave. According to Boyle's Law, when the amount of gas and the temperature stay constant, the pressure and volume of a gas are inversely related. This means that if you increase the volume of the gas, the pressure decreases, and if you decrease the volume, the pressure increases. Mathematically, this relationship can be shown as: \[ P_1 \times V_1 = P_2 \times V_2 \] This equation helps us find out the new volume or pressure of a gas if the other variable changes.
unit conversion
Unit conversion is often necessary when working with gas laws. Different pressures can be given in units like atm, mmHg, or Torr. To use them in formulas, we need to convert these units into one standard unit, usually atmospheres (atm). For example, to convert from mmHg to atm, use the fact that 1 atm = 760 mmHg. So, \[ P_{atm} = \frac{2500 \text{ mmHg}}{760} \text{ mmHg/atm} \ P_{atm} \thickapprox 3.29 \text{atm}\] For converting Torr to atm where 1 atm = 760 Torr, \[ P_{atm} = \frac{80 \text{ Torr}}{760} \text{ Torr/atm} \ P_{atm} \thickapprox 0.105\text{atm}\] Unit conversion simplifies comparing and using different measurements.

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Most popular questions from this chapter

Use the words inspiration and expiration to describe the part of the breathing cycle that occurs as a result of each of the following: a. The diaphragm contracts. b. The volume of the lungs decreases. c. The pressure within the lungs is less than that of the atmosphere.

A scuba diver \(60 \mathrm{ft}\) below the ocean surface inhales \(50.0 \mathrm{~mL}\) of compressed air from a scuba tank at a pressure of 3.00 atm and a temperature of \(8^{\circ} \mathrm{C}\). What is the final pressure of air, in atmospheres, in the lungs when the gas expands to \(150.0 \mathrm{~mL}\) at a body temperature of \(37^{\circ} \mathrm{C},\) if the amount of gas does not change?

Bacteria and viruses are inactivated by temperatures above \(135^{\circ} \mathrm{C}\). An autoclave contains steam at \(1.00 \mathrm{~atm}\) and \(100^{\circ} \mathrm{C}\). What is the pressure, in atmospheres, when the temperature of the steam in the autoclave reaches \(135^{\circ} \mathrm{C},\) if \(V\) and \(n\) do not change?

In 1783 , Jacques Charles launched his first balloon filled with hydrogen gas, which he chose because it was lighter than air. If the balloon had a volume of \(31000 \mathrm{~L}\), how many kilograms of hydrogen were needed to fill the balloon at STP? (8.6)

At a restaurant, a customer chokes on a piece of food. You put your arms around the person's waist and use your fists to push up on the person's abdomen, an action called the Heimlich maneuver. (8.2) a. How would this action change the volume of the chest and lungs? b. Why does it cause the person to expel the food item from the airway?
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